Global optimization: deterministic approaches
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin u.a.
Springer
1993
|
| Edition: | 2., rev. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 657 - 690 |
| Physical Description: | XVI, 698 S. graph. Darst. |
| ISBN: | 3540560947 0387560947 |
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MARC
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| 100 | 1 | |a Horst, Reiner |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Global optimization |b deterministic approaches |c Reiner Horst ; Hoang Tuy |
| 250 | |a 2., rev. ed. | ||
| 264 | 1 | |a Berlin u.a. |b Springer |c 1993 | |
| 300 | |a XVI, 698 S. |b graph. Darst. | ||
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| 500 | |a Literaturverz. S. 657 - 690 | ||
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| 650 | 7 | |a Programmation non linéiare |2 ram | |
| 650 | 7 | |a minimisation concave |2 inriac | |
| 650 | 7 | |a méthode approximation |2 inriac | |
| 650 | 7 | |a méthode branch and bound |2 inriac | |
| 650 | 7 | |a méthode décomposition |2 inriac | |
| 650 | 7 | |a méthode partition |2 inriac | |
| 650 | 7 | |a optimisation globale |2 inriac | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 4 | |a Nonlinear programming | |
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Record in the Search Index
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| adam_text | CONTENTS
PART A: INTRODUCTION AND BASIC TECHNIQUES 1
CHAPTER I. SOME IMPORTANT CLASSES OF GLOBAL
OPTIMIZATION PROBLEMS 3
1. Global Optimization 3
2. Concave Minimization 9
2.1. Definition and Basic Properties 9
2.2. Brief Survey of Direct Applications 12
2.3. Integer Programming and Concave Minimization 14
2.4. Bilinear Programming and Concave Minimization 19
2.5. Complementarity Problems and Concave Minimization 23
2.6. Max—Min Problems and Concave Minimization 25
3. D.C. Programming and Reverse Convex Constraints 26
3.1. D.C. Programming: Basic Properties 26
3.2. D.C. Programming: Applications 32
3.3. Reverse Convex Constraints 36
3.4. Canonical D.C. Programming Problems 39
4. Lipschitzian Optimization and Systems of Equations
and Inequalities 42
4.1. Lipschitzian Optimization 42
4.2. Systems of Equations and Inequalities 46
CHAPTER II. OUTER APPROXIMATION 51
1. Basic Outer Approximation Method 51
2. Outer Approximation by Convex Polyhedral Sets 56
3. Constraint Dropping Strategies 65
4. On Solving the Subproblems (Qk) 68
4.1. Finding an Initial Polytope B1 and its Vertex Set V1 69
4.2. Computing New Vertices and New Extreme Directions 71
4.3. Identifying Redundant Constraints 82
CHAPTER m. CONCAVITY CUTS 85
1. Concept of a Valid Cut 85
2. Valid Cuts in the Degenerate Case 91
3. Convergence of Cutting Procedures 95
4. Concavity Cuts for Handling Reverse Convex Constraints 100
5. A Class of Generalized Concavity Cuts 104
6. Cats Using Negative Edge Extensions 108
CHAPTER IV. BRANCH AND BOUND 111
1. A Prototype Branch and Bound Method 111
2. Finiteness and Convergence Conditions 121
3. Typical Partition Sets and their Refinement 132
3.1. Simplices 132
3.2. Rectangles and Polyhedral Cones 137
4. Lower Bounds 139
4.1. Lipschitzian Optimization 140
4.2. Vertex Minima 141
4.3. Convex Subfunctionals 142
4.4. Duality 153
4.5. Consistency 158
5. Deletion by Infeasibility 163
6. Restart Branch and Bound Algorithm 169
PART B: CONCAVE MINIMIZATION 173
CHAPTER V. CUTTING METHODS 175
1. A Pure Cutting Algorithm 175
1.1. Valid Cuts and a Sufficient Condition
for Global Optimality 176
1.2. Outline of the Method 181
2. Facial Cut Algorithm 184
2.1. The Basic Idea 184
2.2. Finding an Extreme Face of D Relative to M 186
2.3. Facial Valid Cuts 190
2.4. A Finite Cutting Algorithm 192
3. Cut and Split Algorithm 195
3.1. Partition of a Cone 196
3.2. Outline of the Method 197
3.3. Remarks 200
4. Generating Deep Cuts: The Case of Concave
Quadratic Functionals 205
4.1. A Hierarchy of Valid Cuts 205
4.2. Konno s Cutting Method for Concave Quadratic Programming 211
4.3. Bilinear Programming Cuts 216
CHAPTER VI. SUCCESSIVE APPROXIMATION METHODS 219
1. Outer Approximation Algorithms 219
1.1. Linearly Constrained Problem 220
1.2. Problems with Convex Constraints 228
1.3. Reducing the Sizes of the Relaxed Problems 233
2. Inner Approximation 237
2.1. The (DG) Problem 238
2.2. The Concept of Polyhedral Annexation 239
2.3. Computing the Facets of a Polytope 241
2.4. A Polyhedral Annexation Algorithm 244
2.5. Relations to Other Methods 253
2.6. Extensions 256
3. Convex Underestimation 259
3.1. Relaxation and Successive Underestimation 260
3.2. The Falk and Hoffman Algorithm 262
3.3. Rosen s Algorithm 265
4. Concave Polyhedral Underestimation 271
4.1. Outline of the Method 271
4.2. Computation of the Concave Underestimators 273
4.3. Computation of the Nonvertical Facets 274
4.4. Polyhedral Underestimation Algorithm 277
4.5. Alternative Interpretation 279
4.6. Separable Problems 281
CHAPTER VH. SUCCESSIVE PARTITION METHODS 286
1. Conical Algorithms 286
1.1. The Normal Conical Subdivision Process 287
1.2. The Main Subroutine 289
1.3. Construction of Normal Subdivision Processes 291
1.4. The Basic NCS Process 296
1.5. The Normal Conical Algorithm 299
1.6. Remarks Concerning Implementation 303
1.7. Example 306
1.8. Alternative Variants 309
1.9. Concave Minimization with Convex Constraints 314
1.10. Unbounded Feasible Domain 319
1.11. A Class of Exhaustive Subdivision Processes 320
1.12. Exhaustive Nondegenerate Subdivision Processes 326
2. Simplicial Algorithms 333
2.1. Normal Simplicial Subdivision Processes 334
2.2. Normal Simplicial Algorithm 335
2.3. Construction of an NSS Process 337
2.4. The Basic NSS Process 339
2.5. Normal Simplicial Algorithm for Problems with
Convex Constraints 341
3. An Exact Simplicial Algorithm 344
3.1. Simplicial Subdivision of a Polytope 344
3.2. A Finite Branch and Bound Procedure 346
3.3. A Modified ES Algorithm 348
3.4. Unbounded Feasible Set 352
4. Rectangular Algorithms 355
4.1. Normal Rectangular Algorithm 357
4.2. Construction of an NRS Process 359
4.3. Specialization to Concave Quadratic Programming 362
4.4. Example 367
CHAPTER Vm. DECOMPOSITION OP LARGE SCALE PROBLEMS 371
1. Decomposition Framework 372
2. Branch and Bound Approach 374
2.1. Normal Simplicial Algorithm 375
2.2. Normal Rectangular Algorithm 378
2.3. Normal Conical Algorithm 380
3. Polyhedral Underestimation Method 381
3.1. Nonseparable Problems 381
3.2. Separable Problems 383
4. Decomposition by Outer Approximation 390
4.1. Basic Idea 391
4.2. Decomposition Algorithm 392
4.3. An Extension 398
4.4. Outer Approximation Versus Successive Partition 402
4.5. Outer Approximation Combined with Branch and Bound 406
5. Decomposition of Concave Minimization Problems
over Networks 410
5.1. The Minimum Concave Cost Flow Problem 410
5.2. The Single Source Uncapacitated Minimum Concave
Cost Flow Problem (SUCF) 414
5.3. Decomposition Method for (SUCF) 420
5.4. Extension 430
CHAPTER IX. SPECIAL PROBLEMS OF CONCAVE MINIMIZATION 434
1. Bilinear Programming 434
1.1. Basic Properties 435
1.2. Cutting Plane Method 438
1.3. Polyhedral Annexation 443
1.4. Conical Algorithm 445
1.5. Outer Approximation Method 449
2. Complementarity Problems 456
2.1. Basic Properties 457
2.2. Polyhedral Annexation Method for the Linear
Complementarity Problem (LCP) 459
2.3. Conical Algorithm for the (LCP) 462
2.4. Other Global Optimization Approaches to (LCP) 470
2.5. The Concave Complementarity Problem 473
3. Parametric Concave Programming 476
3.1. Basic Properties 478
3.2. Outer Approximation Method for (LRCP) 484
3.3. Methods Based on the Edge Property 487
3.4. Conical Algorithms for (LRCP) 494
PART C: GENERAL NONLINEAR PROBLEMS 503
CHAPTER X. D.C. PROGRAMMING 505
1. Outer Approximation Methods for Solving the Canonical
D.C. Programming Problem 505
1.1. Duality between the Objective and the Constraints 506
1.2. Outer Approximation Algorithms for Canonical D.C. Problems 512
1.3. Outer Approximation for Solving Noncanonical
D.C. Problems 527
2. Branch and Bound Methods 539
3. Solving D.C. Problems by a Sequence of Linear Programs
and Line Searches 544
4. Some Special D.C. Problems and Applications 558
4.1. The Design Centering Problem 558
4.2. The Diamond Cutting Problem 567
4.3. Biconvex Programming and Related Problems 577
CHAPTER XI. LIPSCHITZ AND CONTINUOUS OPTIMIZATION 587
1. Brief Introduction into the Global Minimization
of Univariate Lipschitz Functions 588
1.1. Saw Tooth Covers 588
1.2. Algorithms for Solving the Univariate Lipschitz—Problem 593
2. Branch and Bound Algorithms 600
2.1. Branch and Bound Interpretation of Piyavskii s
Univariate Algorithm 601
2.2. Branch and Bouad Methods for Minimizing a Lipschitz Function
over an n dimensional Rectangle 605
2.3. Branch and Bound Methods for Solving Lipschitz
Optimization Problems with General Constraints 616
2.4. Global Optimization of Concave Functions Subject
to Separable Quadratic Constraints 617
3. Ontei Approximation 629
4. The Relief Indicator Method 638
4.1. Separators for f on D 638
4.2. A Global Optimally Criterion 642
4.3. The Relief Indicator Method 646
References 657
Notation 691
Index 694
|
| any_adam_object | 1 |
| author | Horst, Reiner Tuy, Hoang |
| author_facet | Horst, Reiner Tuy, Hoang |
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| dewey-raw | 519.3 |
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| discipline | Mathematik |
| edition | 2., rev. ed. |
| format | Book |
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| indexdate | 2024-07-09T16:36:18Z |
| institution | BVB |
| isbn | 3540560947 0387560947 |
| language | English |
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| spelling | Horst, Reiner Verfasser aut Global optimization deterministic approaches Reiner Horst ; Hoang Tuy 2., rev. ed. Berlin u.a. Springer 1993 XVI, 698 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 657 - 690 Optimaliseren gtt Optimisation mathématique ram Programmation non linéiare ram minimisation concave inriac méthode approximation inriac méthode branch and bound inriac méthode décomposition inriac méthode partition inriac optimisation globale inriac Mathematical optimization Nonlinear programming Globale Optimierung (DE-588)4140067-7 gnd rswk-swf Globale Optimierung (DE-588)4140067-7 s DE-604 Tuy, Hoang Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003684632&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Horst, Reiner Tuy, Hoang Global optimization deterministic approaches Optimaliseren gtt Optimisation mathématique ram Programmation non linéiare ram minimisation concave inriac méthode approximation inriac méthode branch and bound inriac méthode décomposition inriac méthode partition inriac optimisation globale inriac Mathematical optimization Nonlinear programming Globale Optimierung (DE-588)4140067-7 gnd |
| subject_GND | (DE-588)4140067-7 |
| title | Global optimization deterministic approaches |
| title_auth | Global optimization deterministic approaches |
| title_exact_search | Global optimization deterministic approaches |
| title_full | Global optimization deterministic approaches Reiner Horst ; Hoang Tuy |
| title_fullStr | Global optimization deterministic approaches Reiner Horst ; Hoang Tuy |
| title_full_unstemmed | Global optimization deterministic approaches Reiner Horst ; Hoang Tuy |
| title_short | Global optimization |
| title_sort | global optimization deterministic approaches |
| title_sub | deterministic approaches |
| topic | Optimaliseren gtt Optimisation mathématique ram Programmation non linéiare ram minimisation concave inriac méthode approximation inriac méthode branch and bound inriac méthode décomposition inriac méthode partition inriac optimisation globale inriac Mathematical optimization Nonlinear programming Globale Optimierung (DE-588)4140067-7 gnd |
| topic_facet | Optimaliseren Optimisation mathématique Programmation non linéiare minimisation concave méthode approximation méthode branch and bound méthode décomposition méthode partition optimisation globale Mathematical optimization Nonlinear programming Globale Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003684632&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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